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# Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory

Published:2008-09-01
Printed: Sep 2008
• K. Q. Lan
• G. C. Yang
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## Abstract

The well-known Falkner--Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter involved in the equation. It is known that there exists $\lambda^{*}<0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\lambda\ge \lambda^{*}$ and has no positive solutions for $\lambda<\lambda^{*}$. The known numerical result shows $\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in [-0.4,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner--Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner--Skan equation.
 Keywords: Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutions
 MSC Classifications: 34B16 - Singular nonlinear boundary value problems 34B18 - Positive solutions of nonlinear boundary value problems 34B40 - Boundary value problems on infinite intervals 76D10 - Boundary-layer theory, separation and reattachment, higher-order effects

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